List of mathematical symbols - Basic Knowledge 101

List of mathematical symbols

This is a list of symbols used in all branches ofmathematics to express a formula or to represent aconstant.

A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary

choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "" instead of "=", with the latter representing equality of well-formed formulas. In short,

convention dictates the meaning.

Each symbol is shown both inHTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image usiTnegX.

Contents

Guide Basic symbols Symbols based on equality Symbols that point left or right Brackets Other non-letter symbols Letter-based symbols

Letter modifiers Symbols based on Latin letters Symbols based on Hebrew or Greek letters Variations See also References External links

Guide

This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol (nottehat this article doesn't have the latter two, but they could certainly be added).

There is a Wikibooks guide for using maths in LaTeX,[1] and a comprehensive LaTeX symbol list.[2] It is also possible to check to see if a Unicode code point is available as a LaTeX command, or vice versa.[3] Also note that where there is no LaTeX command natively available for a particular symbol (although there may be options that require adding packages), the symbol could be added via other options, such as setting the document up to support Unicode,[4] and entering the character in a variety of ways (e.g. copying and pasting, keyboard shortcuts, the \unicode{} command[5]) as well as other options[6] and extensive additional information[.7][8]

Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. More advanced meanings are included with some symbols listed here.

Symbols based on equality "=": Symbols derived from or similar to the equal sign, including double-headed arrows. Not surprisingly these symbols are often associated with an

equivalence relation.

Symbols that point left or right:Symbols, such as< and >, that appear to point to one side or anothe.r Brackets: Symbols that are placed on either side of a variable or expression, such a|sx|.

Other non-letter symbols:Symbols that do not fall in any of the other categories. Letter-based symbols:Many mathematical symbols are based on, or closely resemble, a letter in some alphabet. This section includes such symbols, including symbols that resemble upside-down letters. Many letters have conventional meanings in various branches of mathematics and physics. These are not listed here. TSheee also section, below, has several lists of such usages.

Letter modifiers: Symbols that can be placed on or next to any letter to modify the letter's meaning.

Symbols based onLatin letters, including those symbols that resemble or contain anX Symbols based onHebrew or Greek letters e.g. , , , , , , , , . Note: symbols resembling are grouped with "V" under Latin letters.

Variations: Usage in languages written right-to-left

Basic symbols

Symbol

in HTML

+

-

?

? ?

? /

Symbol

in TeX

\pm \mp \times \cdot

\div \surd \sqrt{x}

Name

Read as

Explanation

Category

addition

plus; add

4 + 6 means the sum of4 and 6.

arithmetic

disjoint union

the disjoint union of ...

and ...

A1 + A2 means the disjoint union of sets A1 and A2.

set theory

subtraction

minus; take; subtract

36 - 11 means the subtraction of11 from 36.

arithmetic

negative sign

negative; minus;

the opposite of

-3 means the additive inverse of the number 3.

arithmetic

set-theoretic A - B means the set that contains all complement the elements ofA that are not in B.

minus;

without ( can also be used for set-theoretic

set theory complement as described below.)

plus-minus

plus or minus 6 ? 3 means both 6 + 3 and 6 - 3.

arithmetic

plus-minus 10 ? 2 or equivalently10 ? 20% plus or minus means the range from10 - 2 to measurement 10 + 2.

minus-plus

minus or plus

6 ? (3 5) means 6 + (3 - 5) and 6 - (3 + 5).

arithmetic

multiplication

times; 3 ? 4 or 3 4 means the multiplication multiplied by of 3 by 4.

arithmetic

dot product scalar product

dot

u v means the dot product ofvectors

linear algebra u and v

vector

algebra

cross product

vector

product cross

u ? v means the cross product of vectors u and v

linear algebra

vector

algebra

placeholder A ? means a placeholder for an

(silent)

argument of a function. Indicates the functional nature of an expression

functional without assigning a specific symbol for

analysis an argument.

division (Obelus)

divided by; over

6 ? 3 or 6 / 3 means the division of6 by 3.

arithmetic

quotient group

mod

G / H means the quotient of groupG modulo its subgroup H.

group theory

quotient set

mod

A/~ means the set of all~ equivalence classes in A.

set theory

square root

(radical

symbol)

x means the nonnegative number

the (principal) whose square isx.

square root of

real numbers

complex

square root

the (complex) square root of

If z = r exp(i) is represented inpolar coordinates with - < , then z = r exp(i/2).

complex

numbers

summation

Examples 2 + 7 = 9 A1 = {3, 4, 5, 6} A2 = {7, 8, 9, 10} A1 + A2 = {(3, 1), (4, 1), (5, 1), (6, 1), (7, 2), (8, 2), (9, 2), (10, 2)}

36 - 11 = 25

-(-5) = 5

{1, 2, 4} - {1, 3, 4} = {2} The equation x = 5 ? 4, has two solutions,x = 7 and x = 3. If a = 100 ? 1 mm, then a 99 mm and a 101 mm. cos(x ? y) = cos(x) cos(y) sin(x) sin(y). 7 8 = 56

(1, 2, 5) (3, 4, -1) = 6

i jk (1, 2, 5) ? (3, 4, -1) = 1 2 5 = (-22, 16, -2)

3 4 -1

| ? | 2 ? 4 = 0.5 12 / 4 = 3 {0, a, 2a, b, b + a, b + 2a} / {0, b} = {{0, b}, {a, b + a}, {2a, b + 2a}} If we define ~ by x ~ y x - y , then /~ = {x + n : n , x [0,1)}. 4 = 2

-1 = i

\sum

sum over ...

from ... to ...

means

.

of

calculus

indefinite integral or antiderivative

indefinite integral of

- OR the

antiderivative of

f(x) dx means a function whose derivative is f.

calculus

definite

integral

\int

integral from ... to ... of ...

b a

f(x)

dx

means

the

signed area

between the x-axis and the graph of the

with respect function f between x = a and x = b.

b a

x2

dx

=

b3

- 3

a3

to

calculus

line integral C f ds means the integral off along

line/ path/ curve/ integral of ... along ...

the

r is

cauprvaeraCm, etbarizfa(rti(otn))o|frC'(.t)(I|f

dt, where

the curve

calculus is closed, the symbol may be used

instead, as described below.)

Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regardingGauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the

Contour

symbol would be more appropriate.

integral;

A third related symbol is the closed

closed line volume integral, denoted by the symbol

integral

\oint

contour .

If

C

is

a

Jordan

curve about

0,

then

C

1 z

dz

=

2i.

integral of The contour integral can also frequently

calculus be found with a subscript capital letter

C, C, denoting that a closed loop integral is, in fact, around a contour C,

or sometimes dually appropriately, a

circle C. In representations of Gauss's

Law, a subscript capital S, S, is used

to denote that the integration is over a

closed surface.

...

\ldots

\cdots

ellipsis

and so forth Indicates omitted values from a pattern. 1/2 + 1/4 + 1/8 + 1/16 + = 1

\vdots

everywhere

\ddots

therefore

\therefore

therefore; so;

hence

Sometimes used in proofs before logical consequences.

All humans are mortal. Socrates is a human. Socrates is mortal.

everywhere

because

\because

because; Sometimes used in proofs before

since

reasoning.

11 is prime it has no positive integer factors other than itself and one.

everywhere

!

factorial

factorial

means the product

.

combinatorics

logical negation

The statement !A is true if and only ifA is false.

not

propositional logic

A slash placed through another operator is the same as "!" placed in front.

!(!A) A x y !(x = y)

(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation ?A is preferred.)

The statement ?A is true if and only ifA is false.

?

\neg

logical negation

not

A slash placed through another operator is the same as "?" placed in front.

?(?A) A x y ?(x = y)

~

propositional logic (The symbol ~ has many other uses, so

? or the slash notation is preferred.

Computer scientists will often use! but

this is avoided in mathematical texts).

proportionality

\propto

is proportional to;

varies as

y x means that y = kx for some constant k.

if y = 2x, then y x.

everywhere

infinity

is an element of theextended

\infty

infinity number line that is greater than all real numbers numbers; it often occurs inlimits.

\blacksquare

end of proof

\Box

QED; tombstone;

Used to mark the end of a proof.

Halmos finality symbol

(May also be writtenQ.E.D.)

everywhere \blacktriangleright

Symbols based on equality

Symbol

in HTML

=

~

=:

Symbol

in TeX \ne

\approx

\sim

Name

Read as

Explanation

Examples

Category

equality

is equal to; equals

means and represent the same thing or value.

everywhere

inequality

means that and do not represent the same thing or value.

is not equal to;

does not equal (The forms !=, /= or are generally used in programming

everywhere languages where ease of typing and use ofASCII text is preferred.)

approximately equal

is approximately equal to

x y means x is approximately equal toy. This may also be written, , ~, (Libra Symbol),or .

everywhere

3.14159

isomorphism

G H means that group G is isomorphic (structurally identical) to

is isomorphic to group H.

group theory ( can also be used for isomorphic, as described below.)

Q8 / C2 V

probability distribution

has distribution

X ~ D, means the random variable X has the probability distribution D.

statistics

row equivalence

is row equivalent A ~ B means that B can be generated by using a series of

to

elementary row operationson A

matrix theory

same order of

magnitude

roughly similar; poorly

m ~ n means the quantitiesm and n have the same order of magnitude, or general size.

approximates; is on the order of

(Note that ~ is used for an approximation that is poo,rotherwise use .)

approximation

theory

similarity

is similar to[9]

ABC ~ DEF means triangle ABC is similar to (has the same shape) triangle DEF.

geometry

asymptotically equivalent

is asymptotically equivalent to

f ~ g means

.

asymptotic analysis

equivalence relation

are in the same a ~ b means equivalence class

(and equivalently

).

everywhere

X ~ N(0,1), the standard normal distribution

2 ~ 5 8 ? 9 ~ 100 but 2 10

x ~ x+1 1 ~ 5 mod 4

:=

\equiv

definition

x := y, y =: x or x y means x is defined to be another name fory,

:\Leftrightarrow

is defined as; under certain assumptions taken in context.

:

is equal by definition to

(Some writers use to mean congruence).

\triangleq

everywhere P Q means P is defined to belogically equivalentto Q.

\overset{\underset{\mathrm{def}}

{}}{=}

\doteq

congruence

is congruent to

ABC DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.

geometry

\cong

isomorphic

G H means that group G is isomorphic (structurally identical) to

is isomorphic to group H.

V C2 ? C2

abstract algebra ( can also be used for isomorphic, as described above).

congruence

relation

\equiv

... is congruent to a b (mod n) means a - b is divisible by n ... modulo ...

5 2 (mod 3)

modular arithmetic

\Leftrightarrow

material equivalence

A B means A is true if B is true and A is false if B is false.

x + 5 =y + 2 x + 3 =y

if and only if;

\iff

iff

propositional logic

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